*3.3. Aerodynamic and Total-to-Total E*ffi*ciencies*

Herein, we calculate the aerodynamic and total-to-total efficiencies to evaluate the performance of the gas turbine under various blade conditions. These efficiencies are calculated using Equations (30) and (31):

$$
\eta\_d = \frac{F\_L}{F\_D} \tag{30}
$$

$$\eta\_{\rm tot} = \frac{T\omega}{\dot{m}C\_p T\_0 \left\{1 - \left(P/P\_0\right)^{\frac{\kappa - 1}{\kappa}}\right\}} \,\,\,\tag{31}$$

where η*a* is the aerodynamic efficiency, *FL* is the lift force, *FD* is the drag force, η*tot* is the total-to-total efficiency, *T* is the torque, ω is the angular velocity, .*m* is the mass flow rate, *Cp* is the specific heat of ideal air, κ is the ratio of specific heat, *P* is the outlet mass-averaged total pressure, and *T*0 and *P*0 are the average temperature and total pressure of the turbine inlet, respectively.

Figure 18 presents the aerodynamic and total-to-total efficiencies under various blade conditions. Both the efficiencies are strongly dependent on the damage locations and exhibit the same trends. Compared with the reference case, the top-damage cases exhibited a slight increase in efficiency. Conversely, the middle-damage cases exhibited a significant efficiency reduction. The aerodynamic efficiency is strongly affected by lift and drag forces, while the total-to-total efficiency is significantly affected by torque and outlet pressure. Figure 19 shows the drag force, lift force, torque, and outlet pressure under various blade conditions. Compared with the reference case, the middle-damage cases exhibited a significant increase in drag force, while the top-damage cases exhibited a reduction. In contrast, the middle-damage cases exhibited a decrease in lift force, while the top-damage cases exhibited a slight increase. The changes in the drag and lift forces can be explained by the velocity contour shown in Figure 5a. In the middle-damage cases, the circulation zone on the suction side was closer to the leading edge at the mid-span than in the other cases. This means that the flow separation point in the middle-damage cases was closer to the leading edge than in the other cases. According to flight theory, the closer the separation point moves to the leading edge, the higher the drag and the lower the lift that are generated. In the top-damage cases, the separation point was farther away from the leading edge than in the reference case. Therefore, these cases exhibited higher lift and lower drag forces than the reference case. As a result, the aerodynamic efficiency increased slightly in the top-damage cases but was significantly reduced in the middle-damage cases.

The torque values directly affected the total-to-total efficiency. Similar to the lift force, the torque in the top-damage cases was slightly higher than in the other cases—including the reference case—under normal blade conditions. The outlet pressure also increased in the damaged blades due to the pressure loss at the damage location. The increase in torque and outlet pressure resulted in an improvement in the total-to-total efficiency in the top-damage cases. However, in the middle-damage cases, the decrease in torque was greater than the increase in outlet pressure; for example, in comparison with the reference case, the torque decreased by approximately 2.8% while the outlet pressure only increased by approximately 1.2% in the middle-damage at the pressure side case. Hence, compared with that in the reference case, the total-to-total efficiency in the middle-damage cases were noticeably reduced. Overall, it can be concluded that the efficiency of gas turbines is strongly dependent on the blade

conditions—either normal or damaged. Moreover, if the blades are damaged, the damage locations significantly affect the turbine efficiency. Specifically, if the top part of the blade is damaged, the turbine efficiency can be slightly increased after modification.

**Figure 18.** Total-to-total and aerodynamic efficiencies under various blade conditions.

**Figure 19.** Drag force, lift force, torque, and outlet pressure under various blade conditions.
